Factorise
Question: Factorise $1-27 a^{3}$ Solution: $1-27 a^{3}=1^{3}-(3 a)^{3}$ $=(1-3 a)\left[1^{2}+1 \times 3 x+(3 a)^{2}\right]$ $=(1-3 a)\left(1+3 a+9 a^{2}\right)$...
Read More →A passenger train takes 3 hours less for a journey of 360 km,
Question: A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed? Solution: Let the usual speed of train be $x \mathrm{~km} / \mathrm{hr}$ then, Increased speed of the train $=(x+10) \mathrm{km} / \mathrm{hr}$ Time taken by the train under usual speed to cover $360 \mathrm{~km}=\frac{360}{x} \mathrm{hr}$ Time taken by the train under increased speed to cover $360 \mathrm{~km}=\frac{360}{(x+10)} \mathrm{hr}...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x+2| \leq 9$, then (a) $x \in(-7,11)$ (b) $x \in[-11,7]$ (c) $x \in(-\infty,-7) \cup(11, \infty)$ (d) $x \in(-\infty,-7) \cup[11, \infty)$ Solution: $|x+2| \leq 9$ $\Rightarrow-9 \leq x+2 \leq 9$ $\Rightarrow-9-2 \leq x+2-2 \leq 9-2$ $\Rightarrow-11 \leq x \leq 7$ $\Rightarrow x \in[-11,7]$ Hence, the correct option is (b)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x+2| \leq 9$, then (a) $x \in(-7,11)$ (b) $x \in[-11,7]$ (c) $x \in(-\infty,-7) \cup(11, \infty)$ (d) $x \in(-\infty,-7) \cup[11, \infty)$ Solution: $|x+2| \leq 9$ $\Rightarrow-9 \leq x+2 \leq 9$ $\Rightarrow-9-2 \leq x+2-2 \leq 9-2$ $\Rightarrow-11 \leq x \leq 7$ $\Rightarrow x \in[-11,7]$ Hence, the correct option is (b)....
Read More →Factorize:
Question: Factorize: $a^{3}+0.008$ Solution: $a^{3}+0.008=a^{3}+(0.2)^{3}$ $=(a+0.2)\left[a^{2}-a \times(0.2)+(0.2)^{2}\right]$ $=(a+0.2)\left(a^{2}-0.2 a+0.04\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x-1|5$, then (a) $x \in(-4,6)$ (b) $x \in[-4,6]$ (c) $x \in(-\infty,-4) \cup(6, \infty)$ (d) $x \in(-\infty,-4) \cup[6, \infty)$ Solution: $|x-1|5$ $\Rightarrow x-15$ or $x-1-5$ $\Rightarrow x5+1$ or $x-5+1$ $\Rightarrow x6$ or $x-4$ $\Rightarrow x \in(-\infty,-4) \cup(6, \infty)$ Hence, the correct option is (c)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $|x-1|5$, then (a) $x \in(-4,6)$ (b) $x \in[-4,6]$ (c) $x \in(-\infty,-4) \cup(6, \infty)$ (d) $x \in(-\infty,-4) \cup[6, \infty)$ Solution: $|x-1|5$ $\Rightarrow x-15$ or $x-1-5$ $\Rightarrow x5+1$ or $x-5+1$ $\Rightarrow x6$ or $x-4$ $\Rightarrow x \in(-\infty,-4) \cup(6, \infty)$ Hence, the correct option is (c)....
Read More →Factorize:
Question: Factorize: $x^{5}+x^{2}$ Solution: $x^{5}+x^{2}=x^{2}\left(x^{3}+1\right)$ $=x^{2}\left(x^{3}+1^{3}\right)$ $=x^{2}(x+1)\left(x^{2}-x \times 1+1^{2}\right)$ $=x^{2}(x+1)\left(x^{2}-x+1\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x$ and $a$ are real numbers such that $a0$ and $|x|a$, then (a) $x \in(-a, \infty)$ (b) $x \in[-\infty, a]$ (c) $x \in(-a, a)$ (d) $x \in(-\infty,-a) \cup(a, \infty)$ Solution: If $x$ and $a$ are real numbers such that $a0$. $|x|a$ $\Rightarrow xa$ or $x-a$ $\Rightarrow x \in(-\infty,-a) \cup(a, \infty)$ Hence, the correct option is (d)....
Read More →The speed of a boat in still water is 8 km/hr.
Question: The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream. Solution: Let the speed of stream be $x \mathrm{~km} / \mathrm{hr}$ then Speed downstream $=(8+x) \mathrm{km} / \mathrm{hr}$. Therefore, Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Time taken by the boat to go $15 \mathrm{~km}$ upstream $=\frac{15}{(8-x)} \mathrm{hr}$ Time taken by the boat to returns $22 \mathrm{~km}$ downstream $=\frac{22}{(8+x)...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x$ and $a$ are real numbers such that $a0$ and $|x|a$, then (a) $x \in(-a, \infty)$ (b) $x \in[-\infty, a]$ (c) $x \in(-a, a)$ (d) $x \in(-\infty,-a) \cup(a, \infty)$ Solution: If $x$ and $a$ are real numbers such that $a0$. $|x|a$ $\Rightarrow xa$ or $x-a$ $\Rightarrow x \in(-\infty,-a) \cup(a, \infty)$ Hence, the correct option is (d)....
Read More →Factorize:
Question: Factorize: $7 a^{3}+56 b^{3}$ Solution: $7 a^{3}+56 b^{3}=7\left(a^{3}+8 b^{3}\right)$ $=7\left[(a)^{3}+(2 b)^{3}\right]$ $=7(a+2 b)\left[a^{2}-a \times 2 b+(2 b)^{2}\right]$ $=7(a+2 b)\left(a^{2}-2 a b+4 b^{2}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x$ is a real number and $|x|5$, then (a) $x \geq 5$ (b) $-5x5$ (c) $x \leq-5$ (d) $-5 \leq x \leq 5$ Solution: Ifxis a real number. $|x|5$ $\Rightarrow-5x5$ Hence, the correct option is (b)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x$ is a real number and $|x|5$, then (a) $x \geq 5$ (b) $-5x5$ (c) $x \leq-5$ (d) $-5 \leq x \leq 5$ Solution: Ifxis a real number. $|x|5$ $\Rightarrow-5x5$ Hence, the correct option is (b)....
Read More →Factorize:
Question: Factorize: $16 x^{4}+54 x$ Solution: $16 x^{4}+54 x=2 x\left(8 x^{3}+27\right)$ $=2 x\left[(2 x)^{3}+(3)^{3}\right]$ $=2 x(2 x+3)\left[(2 x)^{2}-2 x \times 3+3^{2}\right]$ $=2 x(2 x+3)\left(4 x^{2}-6 x+9\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: Given that $x, y$ and $b$ are real numbers and $xy, b0$, then (a) $x / by / b$ (b) $x / b \leq y / b$ (c) $x / by / b$ (d) $x / b \geq y / b$ Solution: Given that $x, y$ and $b$ are real numbers and $xy, b0$. Both sides of an inequality can be multiplied or divided by the same positive number. $\therefore \frac{x}{b}\frac{y}{b}$ Hence, the correct option is (a)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: Given that $x, y$ and $b$ are real numbers and $xy, b0$, then (a) $x / by / b$ (b) $x / b \leq y / b$ (c) $x / by / b$ (d) $x / b \geq y / b$ Solution: Given that $x, y$ and $b$ are real numbers and $xy, b0$. Both sides of an inequality can be multiplied or divided by the same positive number. $\therefore \frac{x}{b}\frac{y}{b}$ Hence, the correct option is (a)....
Read More →Factorize:
Question: Factorize: $216 x^{3}+\frac{1}{125}$ Solution: $216 x^{3}+\frac{1}{125}=(6 x)^{3}+\left(\frac{1}{5}\right)^{3}$ $=\left(6 x+\frac{1}{5}\right)\left[(6 x)^{2}-6 x \times \frac{1}{5}+\left(\frac{1}{5}\right)^{2}\right]$ $=\left(6 x+\frac{1}{5}\right)\left(36 x^{2}-\frac{6 x}{5}+\frac{1}{25}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $-3 x+17-13$, then (a) $x \in(10, \infty)$ (b) $x \in[10, \infty)$ (c) $x \in(-\infty, 10]$ (d) $x \in[-10,10)$ Solution: $-3 x+17-13$ Subtracting 17 on both sides, we get $\Rightarrow-3 x+17-17-13-17$ $\Rightarrow-3 x-30$ Dividing $-3$ on both sides, we get $\Rightarrow \frac{-3 x}{-3}\frac{-30}{-3}$ $\Rightarrow x10$ $\Rightarrow x \in(10, \infty)$ Hence, the correct option is (a)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $-3 x+17-13$, then (a) $x \in(10, \infty)$ (b) $x \in[10, \infty)$ (c) $x \in(-\infty, 10]$ (d) $x \in[-10,10)$ Solution: $-3 x+17-13$ Subtracting 17 on both sides, we get $\Rightarrow-3 x+17-17-13-17$ $\Rightarrow-3 x-30$ Dividing $-3$ on both sides, we get $\Rightarrow \frac{-3 x}{-3}\frac{-30}{-3}$ $\Rightarrow x10$ $\Rightarrow x \in(10, \infty)$ Hence, the correct option is (a)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x7$, then (a) $-x-7$ (b) $-x \leq-7$ (c) $-x-7$ (d) $-x \geq-7$ Solution: $x7$ subtracting $x$ on both sides, we get $\Rightarrow x-x7-x$ $\Rightarrow 07-x$ subtracting 7 on both sides, we get $\Rightarrow 0-77-x-7$ $\Rightarrow-7-x$ $\Rightarrow-x-7$ Hence, the correct option is (c)....
Read More →Factorize:
Question: Factorize: $125 a^{3}+\frac{1}{8}$ Solution: $125 a^{3}+\frac{1}{8}=(5 a)^{3}+\left(\frac{1}{2}\right)^{3}$ $=\left(5 a+\frac{1}{2}\right)\left[(5 a)^{2}-5 a \times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\right]$ $=\left(5 a+\frac{1}{2}\right)\left(25 a^{2}-\frac{5 a}{2}+\frac{1}{4}\right)$...
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x7$, then (a) $-x-7$ (b) $-x \leq-7$ (c) $-x-7$ (d) $-x \geq-7$ Solution: $x7$ subtracting $x$ on both sides, we get $\Rightarrow x-x7-x$ $\Rightarrow 07-x$ subtracting 7 on both sides, we get $\Rightarrow 0-77-x-7$ $\Rightarrow-7-x$ $\Rightarrow-x-7$ Hence, the correct option is (c)....
Read More →Mark the correct alternative in each of the following:
Question: Mark the correct alternative in each of the following: If $x7$, then (a) $-x-7$ (b) $-x \leq-7$ (c) $-x-7$ (d) $-x \geq-7$ Solution: $x7$ subtracting $x$ on both sides, we get $\Rightarrow x-x7-x$ $\Rightarrow 07-x$ subtracting 7 on both sides, we get $\Rightarrow 0-77-x-7$ $\Rightarrow-7-x$ $\Rightarrow-x-7$ Hence, the correct option is (c)....
Read More →Factorise
Question: Factorise $27 a^{3}+64 b^{3}$ Solution: We know that $x^{3}+y^{3}=(x+y)\left(x^{2}+y^{2}-x y\right)$ Given: $27 a^{3}+64 b^{3}$ $x=3 a, y=4 b$ $27 a^{3}+64 b^{3}=(3 a+4 b)\left(9 a^{2}+16 b^{2}-12 a b\right)$...
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